HOW TO PROVE THE GIVEN VERTICES FORM  A RHOMBUS

Example :

Examine whether the given points

A(2, -3), B (6, 5) C (-2, 1) and D (-6, -7)

forms a rhombus.

Solution :

Distance Between Two Points (x₁, y₁) and (x₂ , y₂)

√(x₂ - x₁)² + (y₂ - y₁)²

Length of AB :

Here x₁ = 2, y₁ = -3, x₂ = 6  and  y₂ = 5

=  √(6-2)²+(5-(-3))²

=  √4² + 8²

=  √(16+64)

=  √80 units

Length of BC :

Here x₁ = 6, y₁ = 5, x₂ = -2  and  y₂ = 1

=  √(-2-6)² + (1-5)²

=  √(-8)² + (-4)²

=  √64 + 16

=  √80 units

Length of CD :

Here x₁ = -2, y₁ = 1, x₂ = -6  and  y₂ = -7

=  √(-6-(-2))²+(-7-1)²

=  √(-6+2)² + (-8)²

=  √(-4)²+64

=  √(16+64)

=  √80 units

Length of DA :

Here x₁ = -6, y₁ = -7, x₂ = 2  and  y₂ = -3

=  √(2-(-6))² + (-3-(-7))²

=  √(2+6)² + (-3+7)²

=  √(8²+4²)

=  √(64+16)

=  √80 units

Since all sides are equal, it may be a square also. To prove it is rhombus, we can prove any one of the following.

So, the given are not vertices of rhombus.

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